On the Spezialschar of Maass

Let $M_k^{(n)}$ be the space of Siegel modular forms of degree $n$ and even weight $k$. In this paper firstly a certain subspace $\mathsf{Spez}(M_k^{(2n)})$ the Spezialschar of $M_k^{(2n)}$ is introduced. In the setting of the Siegel three-fold it is proven that this Spezialschar is the Maass Spezialschar. Secondly an embedding of $M_k^{(2)}$ into a direct sum $\oplus_{\nu = 0}^{\lfloor \frac{k}{10} \rfloor} \text{Sym}^2 M_{k + 2 \nu}$ is given. This leads to a basic characterization of the Spezialschar property. The results of this paper are directly related to the non-vanishing of certain special values of L-functions related to the Gross-Prasad conjecture. This is illustrated by a significant example in the paper.


Introduction
Hans Maass introduced and applied in a series of papers [Ma79I], [Ma79II] and [Ma79III] the concept of a Spezialschar to prove the Saito-Kurokawa conjecture [Za80]. Let M (4) The second topic of this paper is the characterization of the space of Siegel modular forms of degree two and the corresponding Spezialschar in terms of Taylor coefficients and certain differential operators: here ν ∈ N 0 and M Sym k+2ν = Sym 2 (M k+2ν ). Before we summarize the main results we give an example which also serves as an application. Let F 1 , F 2 , F 3 be a Hecke eigenbasis of the space of Siegel cusp forms S It it conjectured by Gross and Prasad [G-P92] that the coefficients α j , β j , γ j are related to special values of certain automorphic L-functions. Recently the Gross-Prasad conjecture has been proven by Ikeda [Ike05] for the Maass Spezialschar and ν = 0. Moreover we show in this paper that the vanishing at such special values has interesting consequences. We have F j ∈ S Maass 20 if and only if the special value β j is zero. More generally: Theorem 0.2 Let k ∈ N 0 be even. Then we have the embedding For F ∈ S (2) k . Then we have and similarly These two theorems give a transparent explanation of our example from a general point of view.

Acknowledgements:
To be entered later.

Notation
Let Z ∈ C n,n and tr the trace of a matrix then we put e{Z} = e 2πi (tr Z) . For l ∈ Z we define π l = (2πi) l . Let x ∈ R then we use Knuth's notation ⌊x⌋ to denote the greatest integer smaller or equal to x. Let A 2 denote the set of half-integral positive-semidefinite matrices. We parametrize the elements T =

Ultraspherical Differential Operators
Let us start with the notation of the ultraspherical polynomial p k,2ν . Let k and ν be elements of N 0 . Let a and b be elements of a commutative ring. Then we put If we specialize the parameters we have p k,0 (a, b) = 1 and p k,2ν (0, 0) = 0 for ν ∈ N. k the vector space of Siegel modular forms on H n with respect to the full modular group Γ n = Sp n (Z). Moreover let S (n) k denote the subspace of cusp forms. If n = 1 we drop the index to simplify notation. We denote the coordinates of the three-fold H 2 by (τ, z,τ ) for ( τ z zτ ) ∈ H 2 and put q = e{τ }, ξ = e{z} andq = e{τ }. Let d k be the dimension of S k .
Definition 1.1 Let k, ν ∈ N 0 and let k be even. Then we define the ultraspherical differential operator D on the space of holomorphic functions F on H 2 in the following way: In the case ν = 0 we get the pullback F (τ, 0,τ) of F on H × H.
Let us further introduce a related Jacobi differential operator D J,m k,2ν . This is given by exchanging π −1 ∂ ∂τ with m in the definition of the ultraspherical differential operator given in (11). Applying the operator D J,m k,2ν on Jacobiforms Φ ∈ J k,m of weight k and index m on H × C matches with the effect of the operator D 2ν introduced in [E-Z85] ( §3, formula (2)) on Φ.
Since F ∈ M (2) k has a Fourier-Jacobi expansion of the form

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DRAFT --DRAFT --DRAFT --DRAFT --DRAFT --it makes sense to consider D k,2ν with respect to this decomposition in a Fourier-Jacobi expansion Lemma 1.2 Let k, ν ∈ N 0 and let k be even. Then D k,2ν maps M We are now ready to act with the ultraspherical differential operator with respect to its Fourier-Jacobi expansion directly on the Fourier-Jacobi expansion of F in a canonical way where all "coefficients" a F m (τ ) = D J,m k,2ν Φ F m (τ ) are modular forms. This shows us, that if we apply the Peterson slash operator | k+2ν γ here γ ∈ Γ to this function with respect to the variable τ , the function is invariant. The same argument also works for the Fourier-Jacobi expansion with respect to τ . From this we deduce Finally the cuspidal conditions in the lemma also follow from symmetry arguments.
Remark 1.4 There are other possibilities for construction of differential operators as used in this section (see Ibukiyama for a overview [Ibu99]). But since

connection between our approach and the theory developped of Eichler and
Zagier [E-Z85] is so useful we decided to do it this way. We also wanted to introduce the concept of Fourier-Jacobi expansion of differential operators, which is interesting in its own right.

Taylor Expansion Of Siegel Modular Forms
The operators D k,2ν can be seen at this point as somewhat artificial. If we apply D k,2ν to Siegel modular forms F we lose information. For example we know that dimS (2) 20 = 3 and contains a two dimensional subspace of Saito-Kurokawa lifts. Since dimS Sym 20 = 1 we obviously lose informations if we apply D 20,0 . But even worse let F 1 and F 2 be a Hecke eigenbasis of the space of Saito-Kurokwa lifts and F 3 a Hecke eigenform of the orthogonal complement then we have D 20,0 F j = 0 for j = 1, 2, 3. The general case seems to be even worse, since for example dimM On the other hand from an optimistic viewpoint we may find about k pieces D k,2ν F which code all the relevant information needed to characterize the Siegel modular forms F . Paul Garrett in his fundamental papers [Ga84] and [Ga87] introduced the method of calculating pullbacks of modular forms to study automorphic L-functions. We also would like to mention the work of Piatetski-Shapiro, Rallis and Gelbart at this point (see also [GPR87]). And recently Ichino in his paper: Pullbacks of Saito-Kurokawa lifts [Ich05] extended Garrett's ideas in a brilliant way to prove the Gross-Prasad conjecture [G-P92] for Saito-Kurokawa lifts. In the new language we have introduced, it is obvious to consider Garretts pullbacks as the 0−th Taylor coefficients of F around z = 0. Hence it seems to be very lucrative to study also the higher Taylor coefficients and hopefully get some transparent link.
Let k ∈ N 0 be even. Let F ∈ M (2) k and Φ ∈ J k,m . Then we denote by the correponding Taylor expansions with respect to z around z = 0. Here we already used the invariance of F and Φ with respect to the transformation z → (−z) since k is even. Suppose χ 2ν 0 is the first non-vanishing Taylor coefficient,

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DRAFT --DRAFT --DRAFT --DRAFT --DRAFT --then we denote 2ν 0 the vanishing order of the underlying form. If the form is identically zero we define the vanishing order to be ∞. To simplify our notation we introduce normalizing factor Further we put Then a straightforward calculation leads to the following useful formula.
Lemma 2.1 Let k, ν ∈ N 0 and let k be even. Let F ∈ M (2) k . Then we have A similiar formula is valid for Jacobiforms with normalizing factor γ J,m k,2ν = γ k,2ν .
EXAMPLE: It is well known that dim S (2) 10 = 1. Let Φ = Φ 10 ∈ S (2) 10 be normalized in such a way that A Φ (1, 1, 1) = 1. Then it follows from D 10,0 Φ = 0 that A Φ (1, 0, 1) = −2 since dim S Sym 10 = 0. Then Φ has the Taylor expansion We can also express the Taylor coefficients χ F 2ν in terms of the modular forms D k,2ν F . This can be done by inverting the formula (22). Finally we get

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DRAFT --DRAFT --DRAFT --DRAFT --DRAFT --Before we state our first main result about the entropy of the family D k,0 F , D k,2 F , D k,4 F . . . we introduce some further notation.
These spaces will be the target of our next consideration. More precisely we define a linear map from the space of Siegel modular forms of degree 2 into these spaces with remarkable properties.
Theorem 2.3 Let k ∈ N 0 be even. Then we have the linear embedding Since D k,0 S (2) k is cuspidal we have the embedding of S (2) Remark 2.4 It can be deduced from [Hei06] that D k,0 ⊕D k,2 is surjective. Hence for k < 20 we have: • M (2) k is isomorphic to M k for k < 10 and for 10 k < 20 and S (2) k ≃ S k ⊕S k+2 .

Proof:
First of all we recall that we have already shown that D k,0 M k and suppose that D k F is identically zero. Then it would follow from our inversion formula (25) that For such F the general theory of Siegel modular forms of degree 2 says that the special function Φ 10 ∈ S (2) k , which we already studied in one of our examples, divides F in the C−algebra of modular forms. And this is fullfilled at least with a 8 DRAFT --DRAFT --DRAFT --DRAFT --DRAFT --power of ⌊ k 10 ⌋+1 = t k > 0. Hence there exists a Siegel modular form G of weight k − 10 t k . But since this weight is negative and non-trival Siegel modular forms of negative weight do not exist the form G has to be identically zero. Hence we have shown that if D k F ≡ 0 then F ≡ 0. And this proves the statement of the theorem.
Remark 2.5 The number ⌊ k 10 ⌋ in the Theorem is optimal. This follows directly from properties of Φ 10 .
Remark 2.6 Let E 2,1 k (f ) be a Klingen Eisenstein series attached to f ∈ S k . Let E k denote an elliptic Eisenstein series of weight k. Then it can be deduced from [Ga87] Remark 2.7 It would be interesting to have a different proof of the Theorem independent of the special properties of Φ 10 .

The Spezialschar
In this section we first recall some basic facts on the Maass Spezialschar [Za80]. Then we determine the image of the Spezialschar in the space W k for all even weights k. Then finally we introduce a Spezialschar as a certain subspace of the space of Siegel modular forms of degree 2n and weight k. Then we show that in the case n = 1 this Spezialschar coincides with the Maass Spezialschar .

Basics of the Maass Spezialschar
Let J k,m be the space of Jacobi forms of weight k and index m. We denote the subspace of cusp forms with J cusp k,m . Let | k,m the slash operator for Jacobi forms and V l (l ∈ N 0 ) be the operator, which maps J k,m to J k,ml . More precisely, let Φ(τ, z) = c(n, r) q n ξ r ∈ J k,m . Then (Φ | k,m V l )(τ, z) = c * (n, r) q n ξ r with c * (n, r) = a|(n,r,l)  • If we restrict the Saito-Kurokawa lifting to Jacobi cusp forms we get Siegel cusp forms.

Definition 3.1 The lifting V is given by the linear map
• Let Φ ∈ J k,m and l, µ ∈ N 0 . Then we have Here T l is the Hecke operator on the space of elliptic modular forms.

The Diagonal of W k
Let (f j ) be the normalized Hecke eigenbasis of M k . With this notation we introduce the diagonal space and the corresponding cuspidal subspace S D k . Now we are ready to distinguish the Maass Spezialschar in the vector spaces W k and W cusp k .
Theorem 3.4 Let k be a natural even number. Let F be a Siegel modular form of degree two and weight k. Then we have Let F be a cuspform. Then we have here α 0 , α, γ ∈ C.

Proof:
We first show that if F is in the Maass Spezialschar then D k,2ν F is an element of the diagonal space. Let ν ∈ N 0 and Φ F 1 be the first Fourier-Jacobi coefficient of F . Then we have Here we applied the Fourier-Jacobi expansion of the differential operator D k,2ν acting on Siegel modular forms. Then we used the formula (32) to interchange the operators D J,l k,2ν and V l to get be a normalized Hecke eigenbasis of S k+2ν . Let 1 ≤ j 1 , j 2 ≤d k+2ν . Then we have which leads to the desired result It remains to look at the Eisenstein part if ν = 0. Since the space of Eisenstein series has the basis E k and is orthogonal to the functions given in (41) we have proven that the Spezialschar property of F implies that D k F ∈ W D k .

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DRAFT --DRAFT --DRAFT --DRAFT --DRAFT --Now let us assume that F / ∈ M Maass k . Then we show that is an isomorphism, we can assume that is identically zero. Altering F by an element of the Maass Spezialschar does not change the property we have to prove. If D k,0 F / ∈ M D k or D k,2 F / ∈ S D k+2 we are done otherwise we can assume that Then we have the orderF = 2ν 0 4 and k 20, be the Taylor expansion of F with χ F 2ν 0 (τ,τ ) ∈ S k+2ν 0 not identically zero. Let Φ 10 ∈ S (2) 10 be the Siegel cusp form (24) of weight 10 and degree 2. It has the properties that χ Φ 10 0 ≡ 0 and χ Φ 10 2 (τ,τ ) = c ∆(τ ) ∆(τ ) with c = 0. Since orderF = 2ν 0 we also have Φ ν 0 10 F.
This means that there exists a G ∈ S k−10ν 0 such that χ G 0 is non-trivial and Hence we have for the first nontrivial Taylor coefficient of F the formula And the coefficient a 1 (τ ) of q is identically zero. Now let us assume for a moment that χ F 2ν 0 ∈ S D k+2ν 0 . Then we have and the coefficient of q is given by is a basis we have α 1 = . . . = αd k+2ν 0 = 0. But since we assumed that orderF = 13 DRAFT --DRAFT --DRAFT --DRAFT --DRAFT --2ν 0 we have a reductio ad absurdum. Hence we have shown that χ F 2ν 0 / ∈ S k+2ν 0 which proves our theorem. Remark 3.7 Let k be a natural even number. Let F be a Siegel modular form of degree two and weight k. Then we have

The Spezialschar
Let G + Sp n (Q) be the rational symplectic group with positive similitude µ. In the sense of Shimura we attach to Hecke pairs the corresponding Hecke algebras We also would like to mention that in the setting of elliptic modular forms the classical Hecke operator T (p) can be normalized such that it is an element of the full Hecke algebra H 1 , but not of the even one H 1 0 . Let g ∈ G + Sp n (Q) with similitude µ(g). Then we put to obtain an element of Sp n (R). We further extend this to H n .
Definition 3.8 Let T ∈ H n . Then we define Here × is the standard embedding of (Sp n , Sp n ) into Sp 2n .
Now we study the action | k ⋊ ⋉ T on the space of modular forms of degree 2n for all T ∈ H n or T ∈ H n 0 . The first thing we would like to mention is that for F ∈ M Spez M Moreover Spez S This follows from the fact that the Hecke operators are self adjoint and that the space of elliptic modular forms has multiplicity one. To make the operator well-defined we used the embedding H × H into the diagonal of H 2 . We can now interchange the differential operators D k,2ν and the Petersson slash operator | * . This leads to So finally it remains to show that if D k,2ν (F | k ⋊ ⋉ T ) = 0 for all ν ∈ N 0 then it follows F | k ⋊ ⋉ T = 0. By looking at the Taylor expansion of the function  (60)

Proof:
Let F ∈ M (2) k . We proceed as follows. In the proof of Theorem 3.10 it has been shown that (2) k ⇐⇒ (D k,2ν F ) | k+2ν ⋊ ⋉ T = 0 for all T ∈ H 0 and ν ∈ N 0 . (63) (this can also be obtained by following the procedure of the proof of Theorem 3.10).
To verify the equation (62) we show that to being an element of the kernel of the operator | ⋊ ⋉ T (p 2 ) implies already to be an element of the kernel of | ⋊ ⋉ T (p) .
To see this we give a more general proof. Let φ ∈ M Sym k and let φ| k ⋊ ⋉ T (p 2 ) = 0. Let (f j ) be a normalized Hecke eigenbasis of M k . Then we have DRAFT --DRAFT --DRAFT --DRAFT --DRAFT --Let us assume that there exists a α i 0 ,j 0 = 0 with i 0 = j 0 . Let us denote λ l (p 2 ) to be the eigenvalue of f l with respect to the Hecke operator T (p 2 ). Then we have From this follows that λ i 0 (p 2 ) = λ j 0 (p 2 ) for all prime numbers p. It is easy to see at this point that then f i 0 and f j 0 have to be cusp forms. In the setting of cusp forms we can apply a result on multiplicity one for SL 2 of D. Ramakrishnan [Ra00](section 4.1) and other people to obtain f i 0 = f j 0 . Since this is a contradiction we have φ ∈ M D k . In other words we have φ| k ⋊ ⋉ T (p) = 0.